Space-frequency equalization for oversampled received signals

ABSTRACT

Techniques for performing space-frequency equalization and spatial equalization in the frequency domain are described. Space-frequency equalization combines signal components across both space and frequency dimensions whereas spatial equalization combines signal components across space. A receiver obtains input symbols for multiple (M) signal copies from multiple (R) receive antennas and multiple (C) times oversampling, where M is equal to R times C. For space-frequency equalization, the receiver derives equalizer coefficients for the M signal copies, e.g., based on MMSE criterion, filters, the input symbols for the M signal copies with the equalizer coefficients, and combines the filtered symbols for the M signal copies to obtain output symbols. Space-frequency equalization may be used for some frequency bins and spatial equalization may be used for other frequency bins to reduce complexity.

I. CLAIM OF PRIORITY UNDER 35 U.S.C. §119

The present Application for Patent claims priority to ProvisionalApplication Ser. No. 60/676,586, entitled “METHOD AND APPARATUS FORFREQUENCY DOMAIN EQUALIZATION IN WIRELESS COMMUNICATIONS,” filed Apr.28, 2005, assigned to the assignee hereof, and expressly incorporatedherein by reference.

BACKGROUND

I. Field

The present disclosure relates generally to communication, and morespecifically to techniques for performing equalization at a receiver ina communication system.

II. Background

In a communication system, a transmitter typically processes (e.g.,encodes, interleaves, symbol maps, spreads, and scrambles) traffic datato generate a sequence of chips. The transmitter then processes the chipsequence to generate a radio frequency (RF) signal and transmits the RFsignal via a communication channel. The communication channel distortsthe transmitted RF signal with a channel response and further degradesthe signal with noise and interference from other transmitters.

A receiver receives the transmitted RF signal and processes the receivedRF signal to obtain samples. The receiver may perform equalization onthe samples to obtain estimates of the chips sent by the transmitter.The receiver then processes (e.g., descrambles, despreads, demodulates,deinterleaves, and decodes) the chip estimates to obtain decoded data.The equalization performed by the receiver typically has a large impacton the quality of the chip estimates as well as the overall performance.

There is therefore a need in the art for techniques to performequalization in a manner to achieve good performance.

SUMMARY

Techniques for performing space-frequency equalization and spatialequalization in the frequency domain are described herein.Space-frequency equalization combines signal components across bothspace and frequency dimensions whereas spatial equalization combinessignal components across space.

According to an embodiment of the invention, an apparatus is describedwhich includes at least one processor and a memory. The processor(s)derive equalizer coefficients for multiple signal copies (or spectralcopies) from multiple receive antennas and oversampling. Theprocessor(s) then filter input symbols for the multiple signal copieswith the equalizer coefficients to obtain output symbols.

According to another embodiment, a method is provided in which equalizercoefficients are derived for multiple signal copies from multiplereceive antennas and oversampling. Input symbols for the multiple signalcopies are filtered with the equalizer coefficients.

According to yet another embodiment, an apparatus is described whichincludes means for deriving equalizer coefficients for multiple signalcopies from multiple receive antennas and oversampling. The apparatusfurther includes means for filtering input symbols for the multiplesignal copies with the equalizer coefficients.

According to yet another embodiment, an apparatus is described whichincludes at least one processor and a memory. The processor(s) obtaininput symbols for multiple (M) signal copies from multiple (R) receiveantennas and multiple (C) times oversampling, where M is equal to Rtimes C. The processor(s) derive equalizer coefficients for the M signalcopies, filter the input symbols for the M signal copies with theequalizer coefficients, and combine filtered symbols for the M signalcopies to obtain output symbols.

According to yet another embodiment, a method is provided in which inputsymbols are obtained for M signal copies from R receive antennas and Ctimes oversampling. Equalizer coefficients are derived for the M signalcopies. The input symbols for the M signal copies are filtered with theequalizer coefficients. The filtered symbols for the M signal copies arecombined to obtain output symbols.

According to yet another embodiment, an apparatus is described whichincludes means for obtaining input symbols for M signal copies from Rreceive antennas and C times oversampling, means for deriving equalizercoefficients for the M signal copies, means for filtering the inputsymbols for the M signal copies with the equalizer coefficients, andmeans for combining filtered symbols for the M signal copies.

According to yet another embodiment, an apparatus is described whichincludes at least one processor and a memory. The processor(s) implementat least one space-frequency equalizer and at least one spatialequalizer. Each space-frequency equalizer combines signal componentsacross spatial and frequency dimensions. Each spatial equalizer combinessignal components across spatial dimension.

According to yet another embodiment, a method is provided in whichsignal components are combined across spatial and frequency dimensionsfor a first set of at least one frequency bin. Signal components arecombined across spatial dimension for a second set of at least onefrequency bin.

According to yet another embodiment, an apparatus is described whichincludes means for combining signal components across spatial andfrequency dimensions for a first set of at least one frequency bin, andmeans for combining signal components across spatial dimension for asecond set of at least one frequency bin.

Various aspects and embodiments of the invention are described infurther detail below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows two transmitters and a receiver in a communication system.

FIG. 2 shows transmission from a single-antenna transmitter to thereceiver.

FIG. 3 shows a signal flow for a frequency domain equalizer for receivediversity.

FIG. 4A shows spectral plots for two receive antennas with 2×oversampling.

FIG. 4B shows spectral plots for two signal copies for one receiveantenna.

FIG. 5 shows a process to perform space-frequency equalization.

FIG. 6 shows a process to perform equalization with a combination ofspace-frequency equalizers and spatial equalizers.

FIG. 7 shows a block diagram of a multi-antenna transmitter and thereceiver.

DETAILED DESCRIPTION

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments.

For clarity, the following nomenclature is used for much of thedescription below. Time-domain scalars are denoted by lower case textwith index n for sample period, e.g., h(n). Frequency-domain scalars aredenoted by upper case text with index k for frequency bin, e.g., H(k).Vectors are denoted by bolded lower case text, e.g., h, and matrices aredenoted by bolded upper case text, e.g., H.

FIG. 1 shows a communication system 100 with two transmitters 110 x and110 y and a receiver 150. Transmitter 110 x is equipped with a singleantenna 112 x, transmitter 110 y is equipped with multiple (T) antennas112 a through 112 t, and receiver 150 is equipped with multiple (R)antennas 152 a through 152 r. A single-input multiple-output (SIMO)channel is formed by the single antenna at transmitter 110 x and the Rantennas at receiver 150. A multiple-input multiple-output (MIMO)channel is formed by the T antennas at transmitter 110 y and the Rantennas at receiver 150. For both transmitters 110 x and 110 y, asingle-input single-output (SISO) channel exists between eachtransmit/receive antenna pair. The SISO channel may be characterized bya time-domain channel impulse response h(n) or a frequency-domainchannel frequency response H(k).

A time-domain representation may be converted to a frequency-domainrepresentation with a K-point fast Fourier transform (FFT) or a K-pointdiscrete Fourier transform (DFT), which may be expressed as:

$\begin{matrix}{{{H(k)} = {\sum\limits_{n = 1}^{K}{{h(n)} \cdot {\mathbb{e}}^{{- {{j2\pi}{({k - 1})}}}{{({n - 1})}/K}}}}},} & {{Eq}\mspace{14mu}(1)}\end{matrix}$where the “−1” in the exponent is due to indices n and k starting with 1instead of 0.

A frequency-domain representation may be converted to a time-domainrepresentation with a K-point inverse FFT (IFFT) or a K-point inverseDFT (IDFT), which may be expressed as:

$\begin{matrix}{{h(n)} = {\frac{1}{K} \cdot {\sum\limits_{k = 1}^{K}{{H(k)} \cdot {{\mathbb{e}}^{{j2\pi}{({k - 1})}{{({n - 1})}/K}}.}}}}} & {{Eq}\mspace{14mu}(2)}\end{matrix}$

FIG. 2 shows a signal flow 200 for data transmission from single-antennatransmitter 110 x to multi-antenna receiver 150. Receiver 150 utilizesreceive diversity, which is reception of a single data stream withmultiple receive antennas. For simplicity, FIG. 2 shows a case with twoantennas at receiver 150 and two times (2×) oversampling of the receivedsignal from each receive antenna. FIG. 2 shows the use of afractionally-spaced frequency domain equalizer (FDE), which performsequalization in the frequency domain. The term “fractionally-spaced”refers to sampling at a higher rate than the rate required by Nyquistsampling theorem.

Transmitter 110 x processes traffic data and generates transmit chipsx(n′) at chip rate, where n′ is an index for chip period. Thetransmitter may append a cyclic prefix to each block of K/2 transmitchips. The cyclic prefix is a repeated portion of the data block and isused to combat intersymbol interference caused by frequency selectivefading, which is a frequency response that is not flat across the systembandwidth. In an actual system, the transmitter sends the transmit chipsequence to the receiver. For signal flow 200, an upsampler 210 insertsa zero after each transmit chip and generates transmit samples x(n) atsample rate, which is twice the chip rate for 2× oversampling, where nis an index for sample period.

The transmit samples are sent from the single transmit antenna and viathe SIMO channel to the two receive antennas. The SISO channel for thefirst receive antenna is modeled by a channel impulse response of h₁(n)in block 220 a and additive noise of n₁(n) via a summer 224 a. The SISOchannel for the second receive antenna is modeled by a channel impulseresponse of h₂(n) in block 220 b and additive noise of n₂(n) via asummer 224 b. The channel impulse response h_(r)(n) for each receiveantenna r, for r=1, 2, includes the effects of any pulse shaping filterat the transmitter, the propagation channel, any front-end filter at thereceiver, and so on.

Receiver 150 digitizes the received signal from each receive antenna attwice the chip rate and obtains input samples at the sample rate (notshown in FIG. 2). The receiver may remove the cyclic prefix, if any,appended in each data block by the transmitter. The time-domain inputsamples r₁(n) from the first receive antenna are transformed to thefrequency domain with a K-point FFT/DFT by a unit 230 a to obtainfrequency-domain input symbols R₁(k), for k=1, . . . , K. The 2×oversampling results in two copies of the signal spectrum beingavailable for each receiver antenna, as shown in FIG. 4A. The tworedundant signal copies in the oversampled spectrum for each receiveantenna are denoted as a lower copy (L) and an upper copy (U). A signalcopy may also be referred to as a spectral copy or by some otherterminology. The first K/2 input symbols R₁(k), for k=1, . . . , K/2,are denoted as symbols R_(1,L)(k), for k=1, . . . , K/2, for the lowercopy and are provided to an equalizer 240 a. The last K/2 input symbolsR₁(k), for k=K/2+1, . . . , K, are denoted as symbols R_(1,U)(k), fork=1, . . . , K/2, for the upper copy and are provided to an equalizer242 a.

Similarly, the time-domain input samples r₂(n) from the second receiveantenna are transformed to the frequency domain with a K-point FFT/DFTby a unit 230 b to obtain frequency-domain input symbols R₂(k), for k=,. . . , K. The first K/2 input symbols R₂(k), for k=1, . . . , K/2, aredenoted as symbols R_(2,L)(k), for k=1, . . . , K/2, for the lower copyand are provided to an equalizer 240 b. The last K/2 input symbolsR₂(k), for k=K/2+1 . . . , K, are denoted as symbols R_(2,U)(k), fork=1, . . . , K/2, for the upper copy and are provided to an equalizer242 b.

Equalizer 240 a filters its input symbols R_(1,L)(k) with itscoefficients W_(1,L)*(k) and provides filtered symbols Y_(1,L)(k), where“*” denotes a complex conjugate. Equalizer 242 a filters its inputsymbols R_(1,U)(k) with its coefficients W_(1,U)*(k) and providesfiltered symbols Y_(1,U)(k). Equalizer 240 b filters its input symbolsR_(2,L)(k) with its coefficients W_(2,L)*(k) and provides filteredsymbols Y_(2,L)(k). Equalizer 242 b filters its input symbols R_(2,U)(k)with its coefficients W_(1,U)(k) and provides filtered symbolsY_(2,U)(k). A summer 244 a sums the filtered symbols Y_(1,L)(k) andY_(2,L)(k) from equalizers 240 a and 240 b, respectively, and providesoutput symbols Y_(L)(k) for the lower copy. A summer 244 b sums thefiltered symbols Y_(1,U)(k) and Y_(2,U)(k) from equalizers 242 a and 242b, respectively, and provides output symbols Y_(U)(k) for the uppercopy. A unit 250 performs a K-point IFFT/IDFT on output symbols Y_(L)(k)and Y_(U)(k) and provides output samples y(n) at the sample rate. Adownsampler 252 discards every other output sample and provides outputsamples y(n′) at the chip rate. The output samples y(n′) are furtherprocessed to obtain decoded data.

FIG. 3 shows a frequency-domain signal flow 300 for afractionally-spaced FDE for receive diversity. Signal flow 300 isequivalent to signal flow 200 in FIG. 2 and is also for the case withtwo receive antennas and 2× oversampling.

Transmitter 110 x processes traffic data and generates transmit chipsx(n′). In an actual system, the transmitter sends the transmit chipsequence to the receiver and does not perform any FFT/DFT. However, forsignal flow 300, a unit 310 performs a K/2-point FFT/DFT on the transmitchips x(n′) and provides frequency-domain transmit symbols X(k), fork=1, . . . , K/2. The transmit symbols X(k) are sent from the singletransmit antenna and via the SIMO channel to the two receive antennas.The SISO channel for the first receive antenna is modeled by (1) afrequency response of H_(1,L)(k) in block 320 a and additive noise ofN_(1,L)(k) via a summer 324 a for the lower copy and (2) a frequencyresponse of H_(1,U)(k) in block 322 a and additive noise of N_(1,U)(k)via a summer 326 a for the upper copy. A unit 328 a transformstime-domain noise n₁(n) and provides frequency-domain noise N_(1,L)(k)and N_(1,U)(k). Similarly, the SISO channel for the second receiveantenna is modeled by (1) a frequency response of H_(2,L)(k) in block320 b and additive noise of N_(2,L)(k) via a summer 324 b for the lowercopy and (2) a frequency response of H_(2,U)(k) in block 322 b andadditive noise of N_(2,U)(k) via a summer 326 b for the upper copy. Aunit 328 b transforms time-domain noise n₂(n) and providesfrequency-domain noise N_(2,L)(k) and N_(2,U)(k). As shown in FIG. 3,the transmit symbols X(k) are sent via all four blocks 320 a, 320 b, 322a and 322 b.

At receiver 150, an equalizer 340 a receives frequency-domain inputsymbols R_(1,L)(k) from summer 324 a, filters the input symbols with itscoefficients W_(1,L)*(k), and provides filtered symbols Y_(1,L)(k). Anequalizer 342 a receives input symbols R_(1,U)(k) from summer 326 a,filters the input symbols with its coefficients W_(1,U)*(k), andprovides filtered symbols Y_(1,U)(k). An equalizer 340 b receives inputsymbols R_(2,L)(k) from summer 324 b, filters the input symbols with itscoefficients W_(2,L)*(k), and provides filtered symbols Y_(2,L)(k). Anequalizer 342 b receives input symbols R_(2,U)(k) from summer 326 b,filters the input symbols with its coefficients W_(2,U)*(k), andprovides filtered symbols Y_(2,U)(k).

A summer 344 a sums the filtered symbols Y_(1,L)(k) and Y_(1,U)(k) fromequalizers 340 a and 342 a, respectively, and provides filtered symbolsY₁(k) for the first receive antenna. A summer 344 b sums the filteredsymbols Y_(2,L)(k) and Y_(2,U)(k) from equalizers 340 b and 342 b,respectively, and provides filtered symbols Y₂(k) for the second receiveantenna. A summer 346 sums the filtered symbols Y₁(k) and Y₂(k). A gainelement 348 scales the output of summer 346 with a gain of ½ andprovides output symbols Y(k). A unit 350 performs a K/2-point IFFT/IDFTon the output symbols Y(k) and provides time-domain output samples y(n′)at the chip rate.

In comparing signal flows 200 and 300, the 2× upsampling of x(n′) byupsampler 210 in FIG. 2 followed by a K-point FFT/DFT is equivalent toperforming a K/2-point FFT/DFT on x(n′) by unit 310 in FIG. 3 andduplicating X(k) for the lower and upper copies of the oversampledspectrum. The series of operations of adding Y_(1,L)(k) and Y_(2,L)(k)by summer 244 a, adding Y_(1,U)(k) and Y_(2,U)(k) by summer 244 b,performing a K-point IFFT/IDFT by unit 250, and decimation by a factorof two by decimator 252 in FIG. 2 is equivalent to adding Y_(1,L)(k) andY_(1,U)(k) by summer 344 a, adding Y_(2,L)(k) and Y_(2,U)(k) by summer344 b, adding Y₁(k) and Y₂(k) by summer 346, scaling by ½ with unit 348,and performing a K/2-point IFFT/IDFT by unit 350 in FIG. 3. In FIG. 3,summers 344 a and 344 b perform spectral summations and summer 346performs spatial summation. The spectral and spatial summations may alsobe performed in other manners. For example, in FIG. 3, Y_(1,L)(k) andY_(2,L)(k) may be summed to obtain Y_(L)(k) (which corresponds to theoutput of summer 244 a in FIG. 2), Y_(1,U)(k) and Y_(2,U)(k) may besummed to obtain Y_(U)(k) (which corresponds to the output of summer 244b in FIG. 2), and Y_(L)(k) and Y_(U)(k) may be summed and scaled by ½ toobtain Y(k).

FIG. 4A shows exemplary spectral plots for the two receive antennas with2× oversampling. The data chips x(n′) are at the chip rate of f_(c). Thecorresponding spectrum has a single-sided bandwidth of f_(c)/2 or,equivalently, a double-sided bandwidth of f_(c), and a roll-offdetermined by the pulse shaping filter at the transmitter. The receivedsignal from each receive antenna is digitized at the sample rate off_(s), which is twice the chip rate, or f_(s)=2f_(c). For each receiveantenna, the lower copy covers a frequency range of DC to f_(s)/2, whichcorresponds to bin indices k=1 through K/2, and the upper copy covers afrequency range of f_(s)/2 to f_(s), which corresponds to bin indicesk=K/2+1 through K. For simplicity, FIG. 4A shows similar spectral plotsfor the two receive antennas. In general, the spectral plot for eachreceive antenna r has a shape determined by the frequency responseH_(r)(k) for that antenna. The spectral plots for the two receiveantennas may be different if H₁(k) is not equal to H₂(k), which isnormally the case and is exploited for receive diversity.

As shown in FIG. 4A, the receiver obtains four signal copies from aredundancy factor of two from the two receive antennas and anotherredundancy factor of two from the 2× oversampling. FIG. 4A also showshow four redundant signal components in the four signal copies should becombined. The two redundant signal components for each receive antennasare separated by a distance of f_(s)/2 or K/2 frequency bins.

As shown in FIG. 4A, a space-frequency equalizer may be used for eachfrequency bin k, for k=1, . . . , K/2. The space-frequency equalizer forfrequency bin k may combine the redundant signal components on bins kand k+K/2 for both receive antennas. K/2 space-frequency equalizers maybe used for the K/2 frequency bins. For clarity, the processing for onefrequency bin k is described below. The same processing may be performedfor each of the K/2 frequency bins, or for k=1, . . . , K/2.

For a SIMO transmission from transmitter 110 x to receiver 150, thefrequency-domain input symbols at the receiver may be expressed as:r (k)= h (k)·X(k)+ n (k),  Eq (3)where r(k)=[R_(1,L)(k) R_(2,L)(k)R_(1,U)(k) R_(2,U)(k)]^(T) is a 4×1vector of input symbols,

-   -   h(k)=[H_(1,L)(k) H_(2,L)(k) H_(1,U)(k) H_(2,U)(k)]^(T) is a 4×1        vector of channel gains,    -   n(k)=[N_(1,L)(k) N_(2,L)(k)N_(1,U)(k) N_(2,U)(k)]^(T) is a 4×1        vector of noise, and    -   “^(T)” denotes a transpose.        The upper and lower signal copies for each receive antenna are        denoted by subscripts U and L, respectively, and are separated        by K/2 frequency bins, as shown in FIG. 4A.

The frequency-domain output symbols from the FDE may be expressed as:

$\begin{matrix}\begin{matrix}{{{Y(k)} = {{{\underset{\_}{w}}^{H}(k)} \cdot {\underset{\_}{r}(k)}}},} \\{{= {{{{\underset{\_}{w}}^{H}(k)} \cdot {\underset{\_}{h}(k)} \cdot {X(k)}} + {{{\underset{\_}{w}}^{H}(k)} \cdot {\underset{\_}{n}(k)}}}},} \\{{= {{{B(k)} \cdot {X(k)}} + {V(k)}}},}\end{matrix} & {{Eq}\mspace{14mu}(4)}\end{matrix}$where w ^(H)(k)=[W_(1,L)*(k)/2 W_(2,L)*(k)/2 W_(1*U)(k)/2 W_(2,U)*(k)/2]is a 4×1 row vector of equalizer coefficients for frequency bin k,

-   -   B(k)=w ^(H)(k)·h(k) is a scaling for X(k),    -   V(k)=w ^(H)(k)·n(k) is filtered noise for X(k), and    -   “^(h)” denotes a conjugate transpose.        In equation (4), the equalizer coefficients w ^(H)(k) include        the scaling factor of ½ for gain element 348 in FIG. 3.

The equalizer coefficients may be derived based on a minimum mean squareerror (MMSE) technique, a zero-forcing (ZF) technique, a maximal ratiocombining (MRC) technique, and so on. For the MMSE technique, theequalizer coefficients satisfy the following condition:

$\begin{matrix}{{\min\limits_{{\underset{\_}{w}}^{H}{(k)}}{E\left\{ {{{{\underset{\_}{w}}^{H}(k)}{{\cdot {\underset{\_}{r}(k)}} - {X(k)}}}}^{2} \right\}}},} & {{Eq}\mspace{14mu}(5)}\end{matrix}$where E{ } is an expectation operation. Equation (5) minimizes the meansquare error between the FDE output Y(k) and the transmitted symbolsX(k).

The MMSE solution to equation (5) may be expressed as:w ^(H)(k)=S(k)· h ^(H)(k)·[S(k)· h (k)· h ^(H)(k)+ R (k)]⁻¹,  Eq (6)where S(k)=E{|X(k)|²} is the power spectrum of transmit chips x(n′), and

R(k)=E{n(k)·n ^(H)(k)} is a 4×4 noise covariance matrix.

The matrix inversion lemma may be applied to equation (6). The equalizercoefficients may then be expressed as:

$\begin{matrix}{{{\underset{\_}{w}}^{H}(k)} = {\frac{S(k)}{1 + {{S(k)} \cdot {{\underset{\_}{h}}^{H}(k)} \cdot {{\underset{\_}{R}}^{- 1}(k)} \cdot {\underset{\_}{h}(k)}}} \cdot {{\underset{\_}{h}}^{H}(k)} \cdot {{{\underset{\_}{R}}^{- 1}(k)}.}}} & {{Eq}\mspace{14mu}(7)}\end{matrix}$Equation (7) has a 4×4 matrix inversion for R ⁻¹(k) for each frequencybin k. Equation (7) may be simplified as described below.

In a first simplification scheme, which may be used if the lower andupper copies of the oversampled spectrum have uncorrelated noise ornegligible noise correlation, the noise covariance matrix has thefollowing block diagonal form:

$\begin{matrix}{{{{\underset{\_}{R}(k)} = \begin{bmatrix}{{\underset{\_}{R}}_{L}(k)} & 0 \\0 & {{\underset{\_}{R}}_{U}(k)}\end{bmatrix}},{where}}\text{}{{{{\underset{\_}{R}}_{c}(k)} = \begin{bmatrix}{\sigma_{1,c}^{2}(k)} & {{\sigma_{1,c}(k)} \cdot {\sigma_{2,c}(k)} \cdot {\rho_{c}(k)}} \\{{\sigma_{1,c}(k)} \cdot {\sigma_{2,c}(k)} \cdot {\rho_{c}^{*}(k)}} & {\sigma_{2,c}^{2}(k)}\end{bmatrix}};}} & {{Eq}\mspace{14mu}(8)}\end{matrix}$

-   -   σ_(r,c) ²(k)=E{|N_(r,c)(k)|²} is the variance of the noise for        copy c from antenna r,

${\rho_{c}(k)} = \frac{E\left\{ {{N_{1,c}(k)} \cdot {N_{2,c}^{*}(k)}} \right\}}{{\sigma_{1,c}(k)} \cdot {\sigma_{2,c}(k)}}$is the noise correlation between the two receive antennas,

-   -   cε{L, U} is an index for the lower and upper copies, and    -   rε{1,2} is an index for the two receive antennas.

R _(c)(k) is a 2×2 noise covariance matrix for the two receive antennasfor one frequency bin k in signal copy c. The simplification in equation(8) may also be made if the correlation between the two receive antennasis negligible but the noise components in the lower and upper copies arespectrally correlated. In this case, the 4×1 vectors for equation (3)may be reordered to obtain the block diagonal matrix shown in equation(8).

With R(k) defined as shown in equation (8), the equalizer coefficientsin w ^(H)(k) may be expressed as:

$\begin{matrix}{{\frac{W_{r,c}^{*}(k)}{2} = \frac{{S(k)} \cdot \left\lbrack {{{\underset{\_}{h}}_{c}^{H}(k)} \cdot {{\underset{\_}{R}}_{c}^{- 1}(k)}} \right\rbrack_{r}}{D(k)}},{{{for}\mspace{14mu} r} = 1},{{2\mspace{14mu}{and}\mspace{14mu} c} = L},U,} & {{Eq}\mspace{14mu}(9)}\end{matrix}$where h_(c)(k)=[H_(1,c)(k) H_(2,c)(k)]^(T) is a 2×1 vector of channelgains for bin k of copy c,

$\begin{matrix}{{\left\lbrack {{{\underset{\_}{h}}_{c}^{H}(k)} \cdot {{\underset{\_}{R}}_{c}^{- 1}(k)}} \right\rbrack_{1} = \frac{{{H_{1,c}^{*}(k)} \cdot {\sigma_{2,c}^{2}(k)}} - {{H_{2,c}^{*}(k)} \cdot {\rho_{c}^{*}(k)} \cdot {\sigma_{1,c}(k)} \cdot {\sigma_{2,c}(k)}}}{{\sigma_{1,c}^{2}(k)} \cdot {\sigma_{2,c}^{2}(k)} \cdot \left( {1 - {{\rho_{c}(k)}}^{2}} \right)}},} \\{{\left\lbrack {{{\underset{\_}{h}}_{c}^{H}(k)} \cdot {{\underset{\_}{R}}_{c}^{- 1}(k)}} \right\rbrack_{2} = \frac{{{H_{2,c}^{*}(k)} \cdot {\sigma_{1,c}^{2}(k)}} - {{{H_{1,c}^{*}(k)} \cdot \rho_{c}^{*}}{(k) \cdot {\sigma_{1,c}(k)} \cdot {\sigma_{2,c}(k)}}}}{{\sigma_{1,c}^{2}(k)} \cdot {\sigma_{2,c}^{2}(k)} \cdot \left( {1 - {{\rho_{c}(k)}}^{2}} \right)}},{and}} \\{{D(k)} = {1 + {{S(k)} \cdot {\left\lbrack {{{{\underset{\_}{h}}_{L}^{H}(k)} \cdot {{\underset{\_}{R}}_{L}^{- 1}(k)} \cdot {{\underset{\_}{h}}_{L}(k)}} + {{{\underset{\_}{h}}_{U}^{H}(k)} \cdot {{\underset{\_}{R}}_{U}^{- 1}(k)} \cdot {{\underset{\_}{h}}_{U}(k)}}} \right\rbrack.}}}}\end{matrix}$

The components of D(k) may be expanded as follows:

${{{\underset{\_}{h}}_{c}^{H}(k)} \cdot {{\underset{\_}{R}}_{c}^{- 1}(k)} \cdot {{\underset{\_}{h}}_{c}(k)}} = {\frac{{{{H_{1,c}(k)}}^{2}{\sigma_{2,c}^{2}(k)}} + {{{H_{2,c}(k)}}^{2}{\sigma_{1,c}^{2}(k)}} - {2\;{Re}{\left\{ {{H_{1,c}^{*}(k)}{H_{2,c}(k)}{\rho_{c}(k)}} \right\} \cdot {\sigma_{1,c}(k)}}{\sigma_{2,c}(k)}}}{{\sigma_{1,c}^{2}(k)} \cdot {\sigma_{2,c}^{2}(k)} \cdot \left( {1 - {{\rho_{c}(k)}}^{2}} \right)}.}$

In a second simplification scheme, which may be used if the noise isspatially and spectrally uncorrelated and has a spatially and spectrallyequal noise variance, the noise covariance matrix R(k) has the followingform:R (k)=σ²(k)· I,  Eq (10)where σ_(1,L) ²(k)=σ_(2,L) ²(k)=σ_(1,U) ²(k)=σ_(2,U) ²(k)=σ²(k) is thenoise variance,

-   -   ρ_(L)(k)=ρ_(U)(k)=0, and    -   I is an identity matrix.

With R(k) defined as shown in equation (10), the equalizer coefficientsin w ^(H)(k) may be expressed as:

$\begin{matrix}{{\frac{W_{r,c}^{*}(k)}{2} = \frac{{S(k)} \cdot {H_{r,c}^{*}(k)}}{{{S(k)} \cdot {{\underset{\_}{h}(k)}}} + {\sigma^{2}(k)}}},{{{for}\mspace{14mu} r} = 1},{{2\mspace{14mu}{and}{\mspace{11mu}\;}c} = L},U,} & {{Eq}\mspace{14mu}(11)}\end{matrix}$where ∥h(k)∥=|H_(1,L)(k)|²+|H_(2,L)(k)|²+|H_(1,U)(k)|²+|H_(2,U)(k)|².∥h(k)∥ is the norm of the channel response vector h(k) for bin k. Asshown in equation (11), even though the noise is spatially andspectrally uncorrelated, the four equalizer coefficients W_(1,L)*(k)/2,W_(2,L)*(k)/2, W_(1,U)*(k)/2 and W_(2,U)*(k)/2 are jointly determined bythe four spatially and spectrally separated channel gains H_(1,L)(k),H_(2,L)(k), H_(1,U)(k) and H_(2,U)(k).

In a third simplification scheme, which may be used if the noisecomponents for the two receive antenna are uncorrelated so thatρ_(L)(k)=ρ_(U)(k)=0 in equation (8), the noise covariance matrix R(k)has the following form:

$\begin{matrix}{{\underset{\_}{R}(k)} = {\begin{bmatrix}{\sigma_{1,L}^{2}(k)} & 0 & 0 & 0 \\0 & {\sigma_{2,L}^{2}(k)} & 0 & 0 \\0 & 0 & {\sigma_{1,U}^{2}(k)} & 0 \\0 & 0 & 0 & {\sigma_{2,U}^{2}(k)}\end{bmatrix}.}} & {{Eq}\mspace{14mu}(12)}\end{matrix}$For the noise covariance matrix shown in equation (12), different noisevariances may be obtained for different signal copies. The equalizercoefficients w ^(H)(k) may be derived based on R(k) defined as shown inequation (12).

Other simplifications may also be made for other conditions. Forexample, the noise correlation between the two receive antennas may befrequency invariant, so that ρ_(L)(k)=ρ_(L) and ρ_(U)(k)=ρ_(U). Thevarious simplifications may reduce computation for the equalizercoefficients over the computation shown in equation (7).

The signal-to-noise ratio (SNR) for the chip-rate output samples y(n′)may be expressed as:

$\begin{matrix}{{{S\; N\; R_{chip}} = \frac{\sum\limits_{k = 1}^{K/2}\;{S(k)}}{\sum\limits_{k = 1}^{K/2}\;\left\{ {{{{{F \cdot {B(k)}} - 1}}^{2} \cdot {S(k)}} + {{F}^{2} \cdot {\sigma_{V}^{2}(k)}}} \right\}}},} & {{Eq}\mspace{14mu}(13)}\end{matrix}$where σ_(V) ²(k)=E{|V(k)|²}=w ^(H)(k)·R(k)·w(k) is the variance of V(k),

$F = \left\lbrack {\frac{1}{K/2} \cdot {\sum\limits_{k = 1}^{K/2}\;{B(k)}}} \right\rbrack^{- 1}$is a scaling factor, and

SNR_(chip) is the chip SNR for the output samples y(n′).

Equation (4) provides biased MMSE estimates of X(k). The scaling factorF may be applied to either Y(k) or y(n′) to obtain unbiased estimates ofX(k) or x(n′), respectively. If a data symbol is spread across M chipswith a spreading code (e.g., a Walsh code or an OVSF code), then thesymbol SNR for the data symbol may be obtained by multiplying the chipSNR with the spreading code length M.

The fractionally-spaced FDE described above for a SIMO transmission totwo receive antennas may be extended to a SIMO transmission to anynumber of receive antennas. The FDE may also be extended to a MIMOtransmission from multiple (T) transmit antennas to multiple (R) receiveantennas. For clarity, the following description is for a 2×2 MIMOtransmission with two transmit antennas, two receive antennas, and 2×oversampling.

For a MIMO transmission from transmitter 110 y to receiver 150, thefrequency-domain input symbols at the receiver may be expressed as:r (k)= h ₁(k)·X ₁(k)+ h ₂(k)·X ₂(k)+ n (k),  Eq (14)where r(k) is a 4×1 vector of input symbols,

-   -   X₁(k) and X₂(k) are symbols sent from transmit antennas 1 and 2,        respectively,    -   h ₁(k) is a 4×1 vector of channel gains for transmit antenna 1,    -   h ₂(k) is a 4×1 vector of channel gains for transmit antenna 2,        and    -   n(k) is a 4×1 vector of noise.        Vectors r(k), h ₁(k), h ₂(k) and n(k) have the form shown in        equation (3).

Two vectors of equalizer coefficients, w ₁ ^(H)(k) and w ₂ ^(H)(k), maybe derived to recover the two transmitted symbols X₁(k) and X₂(k),respectively, for each frequency bin k. The equalizer coefficientvectors may be derived based on the MMSE, zero-forcing, MRC, or someother technique.

The MMSE equalizer coefficients for each transmit antenna may beexpressed as:

$\begin{matrix}{{{{\underset{\_}{w}}_{t}^{H}(k)} = {\frac{S_{t}(k)}{1 + {{S_{t}(k)} \cdot {{\underset{\_}{h}}_{t}^{H}(k)} \cdot {{\underset{\_}{\Psi}}_{t}^{- 1}(k)} \cdot {{\underset{\_}{h}}_{t}(k)}}} \cdot {{\underset{\_}{h}}_{t}^{H}(k)} \cdot {{\underset{\_}{\Psi}}_{t}^{- 1}(k)}}},{{{for}\mspace{14mu} t} = 1},2,} & {{Eq}\mspace{14mu}(15)}\end{matrix}$where t is an index for the two transmit antennas,

-   -   w _(t) ^(H)(k) is a 1×4 vector of equalizer coefficients for        transmit antenna t,    -   S_(t)(k)=E{|X_(t)(k)|²} is the power spectrum of x_(t)(n) sent        from antenna t, and    -   Ψ _(t)(k) is a 4×4 noise and interference covariance matrix for        transmit antenna t.

The noise and interference covariance matrices for the two transmitantennas may be expressed as:Ψ ₁(k)=S ₂(k)· h ₂(k)· h ₂ ^(H)(k)+ R (k),  and Eq (16)Ψ ₂(k)=S ₁(k)· h ₁(k)· h ₁ ^(H)(k)+ R (k).Equation (16) indicates that the noise and interference covariancematrix Ψ _(t)(k) for transmit antenna t includes (1) the noisecovariance matrix R(k) that is applicable for both transmit antennas and(2) the interference from the data stream sent from the other transmitantenna t, which is S _(t) (k)·h _(t) (k)·h _(t) ^(H)(k). Theinter-stream interference is determined by the channel response vector h_(t) (k) and the power spectrum S _(t) (k) for the other transmitantenna t.

The simplifications described above for the SIMO transmission aregenerally not applicable for the MIMO transmission. This is because Ψ_(t)(k) includes the inter-stream interference from the other transmitantenna. Hence, even if R(k) is a diagonal matrix due to spatially andspectrally uncorrelated noise, the inter-stream interference istypically not a diagonal matrix. Hence, a matrix inversion may beperformed to obtain Ψ _(t) ⁻¹(k) for equation (15)

The equalizer coefficient vectors w ₁ ^(H)(k) and w ₂ ^(H)(k) may beapplied to the input vector r(k) to obtain output symbols Y₁(k) andY₂(k), respectively, which are estimates of the transmitted symbolsX₁(k) and X₂(k), respectively. The frequency-domain output symbols fromthe FDE may be expressed as:Y _(t)(k)= w _(t) ^(H)(k)· r (k)=B _(t)(k)·X _(t)(k)+V _(t)(k), fort=1,2,  Eq (17)where Y_(t)(k) is an estimate of X_(t)(k) sent from transmit antenna t,

-   -   B_(t)(k)=w _(t) ^(H)(k)·h_(t)(k) is a scaling factor for        X_(t)(k), and    -   V_(t)(k)=w _(t) ^(H)(k)·[h _(t) (k)·X _(t) (k)+n(k)] is filtered        noise and interference for X_(t)(k).

The chip SNR for each transmit antenna may be expressed as:

$\begin{matrix}{{{S\; N\; R_{{chip},t}} = \frac{\sum\limits_{k = 1}^{K/2}\;{S_{t}(k)}}{\sum\limits_{k = 1}^{K/2}\;\left\{ {{{{{F_{t} \cdot {B_{t}(k)}} - 1}}^{2} \cdot {S_{t}(k)}} + {{F_{t}}^{2} \cdot {\sigma_{V,t}^{2}(k)}}} \right\}}},} & {{Eq}\mspace{14mu}(18)}\end{matrix}$where σ_(V,t) ²(k)=E{|V_(t)(k)|²}=w _(t) ^(H)(k)·Ψ _(t)(k)·w _(t)(k) isthe variance of V_(t)(k),

$F_{t} = \left\lbrack {\frac{1}{K/2} \cdot {\sum\limits_{k = 1}^{K/2}\;{B_{t}(k)}}} \right\rbrack^{- 1}$is a scaling factor for transmit antenna t, and

SNR_(chip,t) is the chip SNR for transmit antenna t.

Equation (17) provides biased MMSE estimates of X_(t)(k). The scalingfactor F_(t) may be applied to Y_(t)(k) or y_(t)(n′) to obtain unbiasedestimates of X_(t)(k) or x_(t)(n′), respectively. If a data symbol isspread across M chips with a spreading code, then the symbol SNR for thedata symbol may be obtained by multiplying the chip SNR with thespreading code length M.

A space-frequency equalizer structure may be used for a SIMOtransmission or a MIMO transmission. A space-frequency equalizercombines redundant signal components on bins k and k+N/2 for all receiveantennas, as described above. For the case with R=2 and 2× oversampling,a 4×4 matrix inversion for R(k) or Ψ _(t)(k) may be performed for eachequalizer coefficient vector w ^(H)(k) or w _(t) ^(H)(k), respectively.

Referring to FIG. 4A, h _(L)(k)=[H_(1,L)(k) H_(2,L)(k)]^(T) is non-zerofor the passband and the transition band in the lower copy. Similarly, h_(U)(k)=[H_(1,U)(k) H_(2,U)(k)]^(T) is non-zero for the passband and thetransition band in the upper copy. Either h _(L)(k) or h _(U)(k) issmall or zero for frequency bins outside of the passband and transitionband. Hence, for some frequency bins, there are only two redundantsignal components (instead of four) to combine since the signalcomponents at either k or k+K/2 are practically zero.

In an aspect, a combination of space-frequency equalizers and spatialequalizers is used for the K/2 frequency bins in order to reducecomplexity. A space-frequency equalizer may be used for each frequencybin k in which there are non-negligible signal components on both bin kin the lower copy and bin k+K/2 in the upper copy. A spatial equalizermay be used for each frequency bin k in which there are signalcomponents on only bin k in the lower copy or bin k+K/2 in the uppercopy.

FIG. 4B shows a spectral plot of two signal copies for one receiveantenna. As shown in FIG. 4B, for each of frequency bins 1≦k≦K_(A), thesignal component on bin k+K/2 in the upper copy is small or zero, and aspatial equalizer may be used for each of these bins. For each offrequency bins K_(A)<k≦K_(B), the signal components on both bin k in thelower copy and bin k+K/2 in the upper copy are non-negligible, and aspace-frequency equalizer may be used for each of these bins. For eachof frequency bins K_(B)<k≦K/2, the signal component on bin k in thelower copy is small or zero, and a spatial equalizer may be used foreach of these bins. A 4×4 matrix inversion may be performed for eachspace-frequency equalizer if the simplifications described above are notapplicable. A 2×2 matrix inversion may be performed for each spatialequalizer if the simplifications are not applicable. The use of bothspace-frequency equalizers and spatial equalizers generally reducescomplexity without degradation in performance.

For a general case with T transmit antennas and R receive antennas, thechannel response vectors for each transmit antenna t may be defined asfollows:h _(t,L)(k)=[H _(t,1,L)(k)H _(t,2,L)(k) . . . H _(t,R,L)(k)]^(T) is anR×1 vector,h _(t,U)(k)=[H _(t,1,U)(k)H _(t,2,U)(k) . . . H _(t,R,U)(k)]^(T) is anR×1 vector, andh _(t)(k)=[ h _(t,L) ^(T)(k) h _(t,U) ^(T)(k)]^(T) is a 2R×1 vector.

The noise vectors for the R receive antennas may be defined as follows:n _(L)(k)=[N _(1,L)(k)N _(2,L)(k) . . . N _(R,L)(k)]^(T) is an R×1vector,n _(U)(k)=[N _(1,U)(k)N _(2,U)(k) . . . N _(R,U)(k)]^(T) is an R×1vector, andn (k)=[n _(L) ^(T)(k) n _(U) ^(T)(k)]^(T) is a 2R×1 vector.

The noise covariance matrices may be defined as follows:R _(L)(k)=E{n _(L)(k)·n _(L) ^(H)(k)} is an R×R matrix,R _(U)(k)=E{n _(U)(k)·n _(U) ^(H)(k)} is an R×R matrix, andR (k)=E{n (k)· n ^(H)(k)} is a 2R×2R matrix.

The noise and interference covariance matrices may be defined asfollows:

$\begin{matrix}{{{{\underset{\_}{\Psi}}_{t,L}(k)} = {{\sum\limits_{{i = 1},{i \neq t}}^{T}\;{{S_{i}(k)} \cdot {{\underset{\_}{h}}_{i,L}(k)} \cdot {{\underset{\_}{h}}_{i,L}^{H}(k)}}} + {{{\underset{\_}{R}}_{L}(k)}{\mspace{11mu}\;}{is}\mspace{14mu}{an}\mspace{14mu} R \times R\mspace{14mu}{matrix}}}},} \\{{{\underset{\_}{\Psi}}_{t,U}(k)} = {{~~}\;}\begin{matrix}{{\sum\limits_{{i = 1},{i \neq t}}^{T}\;{{S_{i}(k)} \cdot {{\underset{\_}{h}}_{i,U}(k)} \cdot {{\underset{\_}{h}}_{i,U}^{H}(k)}}} + {{\underset{\_}{R}}_{U}(k)}} \\{{{is}\mspace{14mu}{an}\mspace{14mu} R \times R\mspace{14mu}{matrix}},{and}}\end{matrix}} \\{{{{\underset{\_}{\Psi}}_{t}(k)} = {{\sum\limits_{{i = 1},{i \neq t}}^{T}\;{{S_{i}(k)} \cdot {{\underset{\_}{h}}_{i}(k)} \cdot {{\underset{\_}{h}}_{i}^{H}(k)}}} + {{\underset{\_}{R}(k)}\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu} 2\; R \times 2\; R\mspace{14mu}{{matrix}.}}}}\mspace{14mu}}\end{matrix}$

The spatial equalizer for each of frequency bins 1 through K_(A) may beexpressed as:

$\begin{matrix}{{{\underset{\_}{w}}_{{sp},t}^{H}(k)} = {\frac{S_{t}(k)}{1 + {{S_{t}(k)} \cdot {{\underset{\_}{h}}_{t,L}^{H}(k)} \cdot {{\underset{\_}{\Psi}}_{t,L}^{- 1}(k)} \cdot {{\underset{\_}{h}}_{t,L}(k)}}} \cdot {{\underset{\_}{h}}_{t,L}^{H}(k)} \cdot {{{\underset{\_}{\Psi}}_{t,L}^{- 1}(k)}.}}} & {{Eq}\mspace{14mu}(19)}\end{matrix}$

The space-frequency equalizer for each of frequency bins K_(A)+1 throughK_(B) may be expressed as:

$\begin{matrix}{{{\underset{\_}{w}}_{{sf},t}^{H}(k)} = {\frac{S_{t}(k)}{1 + {{S_{t}(k)} \cdot {{\underset{\_}{h}}_{t}^{H}(k)} \cdot {{\underset{\_}{\Psi}}_{t}^{- 1}(k)} \cdot {{\underset{\_}{h}}_{t}(k)}}} \cdot {{\underset{\_}{h}}_{t}^{H}(k)} \cdot {{{\underset{\_}{\Psi}}_{t}^{- 1}(k)}.}}} & {{Eq}\mspace{14mu}(20)}\end{matrix}$

The spatial equalizer for each of frequency bins K_(B)+1 through K/2 maybe expressed as:

$\begin{matrix}{{{\underset{\_}{w}}_{{sp},t}^{H}(k)} = {\frac{S_{t}(k)}{1 + {{S_{t}(k)} \cdot {{\underset{\_}{h}}_{t,U}^{H}(k)} \cdot {{\underset{\_}{\Psi}}_{t,U}^{- 1}(k)} \cdot {{\underset{\_}{h}}_{t,U}(k)}}} \cdot {{\underset{\_}{h}}_{t,U}^{H}(k)} \cdot {{{\underset{\_}{\Psi}}_{t,U}^{- 1}(k)}.}}} & {{Eq}\mspace{14mu}(21)}\end{matrix}$

The space-frequency equalizer coefficients w _(sf,t) ^(H)(k) in equation(20) may be obtained with a 2R×2R matrix inversion. The spatialequalizer coefficients w _(sp,t) ^(H)(k) in equation (19) or (21) may beobtained with an R×R matrix inversion. For R=2, the spatial equalizercoefficients may be derived based on a closed form solution instead of a2×2 matrix inversion.

To further reduce complexity, a common noise and interference covariancematrix may be used for all T transmit antennas by applying the matrixinversion lemma. The equalizer coefficients in equations (19), (20) and(21) may then be expressed as:

$\begin{matrix}\begin{matrix}{{{{\underset{\_}{w}}_{{sp},t}^{H}(k)} = {{S_{t}(k)} \cdot {{\underset{\_}{h}}_{t,L}^{H}(k)} \cdot {{\underset{\_}{\Psi}}_{L}^{- 1}(k)}}},} & {{{{for}\mspace{14mu} 1} \leq k \leq K_{A}},}\end{matrix} & {{Eq}\mspace{14mu}(22)} \\\begin{matrix}{{{{\underset{\_}{w}}_{{sf},t}^{H}(k)} = {{S_{t}(k)} \cdot {{\underset{\_}{h}}_{t}^{H}(k)} \cdot {{\underset{\_}{\Psi}}^{- 1}(k)}}},} & {{{{for}\mspace{14mu} K_{A}} < k \leq K_{B}},{and}}\end{matrix} & {{Eq}\mspace{14mu}(23)} \\\begin{matrix}{{{{\underset{\_}{w}}_{{sp},t}^{H}(k)} = {{S_{t}(k)} \cdot {{\underset{\_}{h}}_{t,U}^{H}(k)} \cdot {{\underset{\_}{\Psi}}_{U}^{- 1}(k)}}},} & {{{{for}\mspace{14mu} K_{B}} < k \leq {K/2}},}\end{matrix} & {{Eq}\mspace{14mu}(24)} \\{{{{where}\mspace{11mu}{\underset{\_}{\;\Psi}}_{L}(k)} = {{\sum\limits_{i = 1}^{T}\;{{S_{i}(k)} \cdot {{\underset{\_}{h}}_{i,L}(k)} \cdot {{\underset{\_}{h}}_{i,L}^{H}(k)}}} + {{\underset{\_}{R}}_{L}(k)}}},} & \; \\{{{{\underset{\_}{\Psi}}_{U}(k)} = {{\sum\limits_{i = 1}^{T}\;{{S_{i}(k)} \cdot {{\underset{\_}{h}}_{i,U}(k)} \cdot {{\underset{\_}{h}}_{{i,U}\;}^{H}(k)}}} + {{\underset{\_}{R}}_{U}(k)}}},{and}} & \; \\{{\underset{\_}{\Psi}(k)} = {{\sum\limits_{i = 1}^{T}\;{{S_{i}(k)} \cdot {{\underset{\_}{h}}_{i}(k)} \cdot {{\underset{\_}{h}}_{i\;}^{H}(k)}}} + {{\underset{\_}{R}(k)}.}}} & \;\end{matrix}$

In an embodiment, frequency bins K_(A) and K_(B) may be defined as:K _(A)=(1−α+ε)·K/4 and  Eq (25)K _(B)=(1+α−ε)·K/4,  Eq (26)where α is a roll-off factor for the pulse shaping filter at thetransmitter, and

ε is an equalizer selection threshold.

The roll-off factor may be specified by the system, e.g., α=0.22 forW-CDMA. The threshold ε determines whether to use space-frequencyequalization or spatial equalization and may be defined as 0≦ε≦α. Withε=0, space-frequency equalizers are used for α·K/2 frequency bins,spatial equalizers are used for the remaining (1−α)·K/2 frequency bins,and significant reduction in complexity may be achieved withoutdegradation in performance. As threshold ε increases, spatial equalizersare used for more frequency bins, complexity further reduces, butperformance may start to degrade. Threshold ε may be selected based on atradeoff between complexity and performance.

FIG. 5 shows a process 500 for performing space-frequency equalization.Frequency-domain input symbols are obtained for multiple (M) signalcopies from multiple (R) receive antennas and multiple (C) timesoversampling, or C signal copies from each receive antenna, where M=R·C(block 512). The input symbols for the M signal copies may be obtainedby (1) receiving time-domain input samples at C times chip rate for eachreceive antenna and (2) transforming the input samples for each receiveantenna to the frequency domain to obtain input symbols for the C signalcopies for the receive antenna.

Equalizer coefficients for the M signal copies are derived, e.g., basedon channel and noise estimates and in accordance with the MMSE criterion(block 514). The input symbols for the M signal copies are filtered withthe equalizer coefficients (block 516). The filtered symbols for the Msignal copies are combined to obtain output symbols (block 518). Msignal components in frequency bin k for the M signal copies may becombined, where k is an index for the K/C frequency bins in each signalcopy.

If one data stream is being recovered for a SIMO transmission, then oneset of equalizer coefficients W_(r,c)*(k) may be derived for each signalcopy. For example, if C=2 and R=2, then four sets of equalizercoefficients W_(1,L)*(k), W_(1,U)*(k), W_(2,L)*(k) and W_(2,U)*(k) maybe derived for four signal copies. For the embodiments described above,each set includes K/2 equalizer coefficients for K/2 frequency bins inone signal copy. A vector of M equalizer coefficients, w ^(H)(k), may beformed for each frequency bin k with M equalizer coefficients forfrequency bin k. The equalizer coefficients may be derived based on anassumption of (1) spectrally uncorrelated noise for the C signal copiesfrom each receive antenna, (2) spatially uncorrelated noise for the Rreceive antennas, or (3) spatially and spectrally uncorrelated noise forthe M signal copies. The computation for the equalizer coefficients maybe simplified with any of the noise assumptions, as described above.

If multiple (T) data streams are being recovered for a MIMOtransmission, then M sets of equalizer coefficients may be derived forthe M signal copies for each data stream. For each frequency bin k, anoise and interference covariance matrix Ψ _(t)(k) may be determined foreach data stream and used to derive the equalizer coefficients w _(t)^(H)(k) for that data stream. Alternatively, for each frequency bin k, acommon noise and interference covariance matrix Ψ(k) may be determined,and the equalizer coefficients for all T data streams may be derivedbased on this common noise and interference covariance matrix. The inputsymbols for the M signal copies may be filtered with the M sets ofequalizer coefficients for each data stream to obtain filtered symbolsfor the M signal copies for the data stream. The filtered symbols forthe M signal copies for each data stream may be combined to obtainoutput symbols for the data stream.

In practice, even when C is greater than 2, the receiver typically doesnot have to combine all M=C·R signal components since in most cases only2R out of M signal components have non-negligible signal energy. All ofthe other redundant components are typically suppressed by the stop-bandof the transmitter filters and the receiver front-end filters.Therefore, the practical dimension of the space-frequency equalizer orthe covariance matrix remains 2R even when C>2.

FIG. 6 shows a process 600 for performing equalization with acombination of space-frequency equalizers and spatial equalizers.Space-frequency equalization is performed for a first set of frequencybins, e.g., frequency bins K_(A)+1 through K_(B) in FIG. 4B (block 612).The space-frequency equalization combines signal components acrossspatial and frequency dimensions. Spatial equalization is performed fora second set of frequency bins, e.g., frequency bins 1 through K_(A) andfrequency bins K_(B)+1 through K/2 in FIG. 4B (block 614). The spatialequalization combines signal components across spatial dimension. Thefirst and second sets of frequency bins may be selected based on thefrequency response of the transmit pulse shaping filter, a tradeoffbetween complexity and performance, and so on (block 616).

FIG. 7 shows a block diagram of transmitter 110 y and receiver 150 insystem 100 in FIG. 1. For a downlink/forward link transmission,transmitter 110 y is part of a base station, and receiver 150 is part ofa wireless device. For an uplink/reverse link transmission, transmitter110 y is part of a wireless device, and receiver 150 is part of a basestation. A base station is typically a fixed station that communicateswith the wireless devices and may also be called a Node B, an accesspoint, and so on. A wireless device may be fixed or mobile and may alsobe called a user equipment (UE), a mobile station, a user terminal, asubscriber unit, and so on. A wireless device may be a cellular phone, apersonal digital assistant (PDA), a wireless modem card, or some otherdevice or apparatus.

At transmitter 110 y, a transmit (TX) data processor 720 processes(e.g., encodes, interleaves, and symbol maps) traffic data and providesdata symbols to T modulators 730 a through 730 t. As used herein, a datasymbol is a modulation symbol for data, a pilot symbol is a modulationsymbol for pilot, a modulation symbol is a complex value for a point ina signal constellation (e.g., for M-PSK or M-QAM), and pilot is datathat is known a priori by both the transmitter and receiver. Eachmodulator 730 processes its data symbols and pilot symbols in the mannerspecified by the system and provides transmit chips x_(t)(n) to anassociated transmitter unit (TMTR) 736. Each transmitter unit 736processes (e.g., converts to analog, amplifies, filters, and frequencyupconverts) its transmit chips and generates a modulated signal. Tmodulated signals from T transmitter units 736 a through 736 t aretransmitted from T antennas 112 a through 112 t, respectively.

At receiver 150, R antennas 152 a through 152 r receive the transmittedsignals via various signal paths and provide R received signals to Rreceiver units (RCVR) 754 a through 754 r, respectively. Each receiverunit 754 conditions (e.g., filters, amplifies, and frequencydownconverts) its received signal, digitizes the conditioned signal atmultiple times (e.g., twice) the chip rate, and provides time-domaininput samples to an associated FFT/DFT unit 756. Each unit 756transforms the input samples to the frequency domain and providesfrequency-domain input symbols R_(r)(k).

A channel and noise estimator 758 may estimate the channel responsevectors and the noise based on the frequency-domain input symbols fromFFT/DFT units 756 (as shown in FIG. 7) and/or the time-domain inputsamples from receiver units 754 (not shown in FIG. 7). Channel and noiseestimation may be performed in various manners known in the art. Afrequency domain equalizer (FDE) 760 derives equalizer coefficientsbased on the channel response vectors and the noise estimates, filtersthe input symbols with the equalizer coefficients, combines the filteredsymbols across space and frequency or just space, and provides outputsymbols to T demodulators (Demod) 770 a through 770 t. Each demodulator770 may perform FFT/IDFT on the output symbols from FDE 760 iftransmitter 110 sends modulation symbols in the time domain, e.g., forCDMA, TDMA, and SC-FDMA. Each demodulator 770 then processes its(frequency or time-domain) output symbols in a manner complementary tothe processing by modulator 730 and provides data symbol estimates. Areceive (RX) data processor 780 processes (e.g., symbol demaps,deinterleaves, and decodes) the data symbol estimates and providesdecoded data. In general, the processing by demodulators 770 and RX dataprocessor 780 is complementary to the processing by modulators 730 andTX data processor 720, respectively, at transmitter 110 y.

Controllers/processors 740 and 790 direct operation of variousprocessing units at transmitter 110 y and receiver 150, respectively.Memories 742 and 792 store data and program codes for transmitter 110 yand receiver 150, respectively.

The equalization techniques described herein may be used for variouscommunication systems such as Code Division Multiple Access (CDMA)systems, Time Division Multiple Access (TDMA) systems, FrequencyDivision Multiple Access (FDMA) systems, Orthogonal Frequency DivisionMultiple Access (OFDMA) systems, Single-Carrier FDMA (SC-FDMA) systems,and so on. A CDMA system may implement one or more radio technologiessuch as Wideband-CDMA (W-CDMA), cdma2000, and so on. cdma2000 coversIS-2000, IS-856, and IS-95 standards. A TDMA system may implement aradio technology such as Global System for Mobile Communications (GSM).These various radio technologies and standards are known in the art.W-CDMA and GSM are described in documents from a consortium named “3rdGeneration Partnership Project” (3GPP). cdma2000 is described indocuments from a consortium named “3rd Generation Partnership Project 2”(3GPP2). 3GPP and 3GPP2 documents are publicly available. An OFDMAsystem transmits modulation symbols in the frequency domain onorthogonal frequency subbands using orthogonal frequency divisionmultiplexing (OFDM). An SC-FDMA system transmits modulation symbols inthe time domain on orthogonal frequency subbands.

Modulators 730 at transmitter 110 y and demodulators 770 at receiver 150perform processing as specified by the system. For example, modulators720 may perform processing for CDMA, OFDM, SC-FDMA, and so on, or acombination thereof.

Those of skill in the art would understand that information and signalsmay be represented using any of a variety of different technologies andtechniques. For example, data, instructions, commands, information,signals, bits, symbols, and chips that may be referenced throughout theabove description may be represented by voltages, currents,electromagnetic waves, magnetic fields or particles, optical fields orparticles, or any combination thereof.

Those of skill would further appreciate that the various illustrativelogical blocks, modules, circuits, and algorithm steps described inconnection with the embodiments disclosed herein may be implemented aselectronic hardware, computer software, or combinations of both. Toclearly illustrate this interchangeability of hardware and software,various illustrative components, blocks, modules, circuits, and stepshave been described above generally in terms of their functionality.Whether such functionality is implemented as hardware or softwaredepends upon the particular application and design constraints imposedon the overall system. Skilled artisans may implement the describedfunctionality in varying ways for each particular application, but suchimplementation decisions should not be interpreted as causing adeparture from the scope of the present invention.

The various illustrative logical blocks, modules, and circuits describedin connection with the embodiments disclosed herein may be implementedor performed with a general purpose processor, a digital signalprocessor (DSP), an application specific integrated circuit (ASIC), afield programmable gate array (FPGA) or other programmable logic device,discrete gate or transistor logic, discrete hardware components, or anycombination thereof designed to perform the functions described herein.A general purpose processor may be a microprocessor, but in thealternative, the processor may be any conventional processor,controller, microcontroller, or state machine. A processor may also beimplemented as a combination of computing devices, e.g., a combinationof a DSP and a microprocessor, a plurality of microprocessors, one ormore microprocessors in conjunction with a DSP core, or any other suchconfiguration.

The steps of a method or algorithm described in connection with theembodiments disclosed herein may be embodied directly in hardware, in asoftware module executed by a processor, or in a combination of the two.A software module may reside in RAM memory, flash memory, ROM memory,EPROM memory, EEPROM memory, registers, hard disk, a removable disk, aCD-ROM, or any other form of storage medium known in the art. Anexemplary storage medium is coupled to the processor such that theprocessor can read information from, and write information to, thestorage medium. In the alternative, the storage medium may be integralto the processor. The processor and the storage medium may reside in anASIC. The ASIC may reside in a user terminal. In the alternative, theprocessor and the storage medium may reside as discrete components in auser terminal.

The previous description of the disclosed embodiments is provided toenable any person skilled in the art to make or use the presentinvention. Various modifications to these embodiments will be readilyapparent to those skilled in the art, and the generic principles definedherein may be applied to other embodiments without departing from thespirit or scope of the invention. Thus, the present invention is notintended to be limited to the embodiments shown herein but is to beaccorded the widest scope consistent with the principles and novelfeatures disclosed herein.

1. An apparatus comprising: at least one processor configured toimplement at least one space-frequency equalizer, each space-frequencyequalizer combining signal components across spatial and frequencydimensions, and to implement at least one spatial equalizer, eachspatial equalizer combining signal components across spatial dimension,wherein the at least one processor implements a set of space-frequencyequalizers for a first set of frequency bins in a received signal andimplements a set of spatial equalizers for a different second set offrequency bins in the received signal; and a memory coupled to the atleast one processor.
 2. The apparatus of claim 1, wherein the first andsecond sets of frequency bins are determined based on a frequencyresponse of a transmit pulse shaping filter.
 3. The apparatus of claim1, wherein the at least one processor derives coefficients for the atleast one space-frequency equalizer and the at least one spatialequalizer based on minimum mean square error (MMSE) criterion.
 4. Amethod comprising: combining signal components across spatial andfrequency dimensions for at least a first frequency bin k₁ of a givencopy of a received signal; and combining signal components acrossspatial dimension for at least a disparate second frequency bin k₂ ofthe given copy of the received signal, where k₁ and k₂ are indexes forfrequency bins.
 5. The method of claim 4, wherein the combining thesignal components across the spatial and frequency dimensions comprisesderiving equalizer coefficients based on minimum mean square error(MMSE) criterion, and filtering input symbols for multiple signal copieswith the equalizer coefficients.
 6. An apparatus comprising: means forcombining signal components across spatial and frequency dimensions forat least a first frequency bin k₁ of a given copy of a received signal;and means for combining signal components across spatial dimension forat least a disparate second frequency bin k₂ of the given copy of thereceived signal, where k₁ and k₂ are indexes for frequency bins.
 7. Theapparatus of claim 6, wherein the means for combining the signalcomponents across the spatial and frequency dimensions comprises: meansfor deriving equalizer coefficients based on minimum mean square error(MMSE) criterion, and means for filtering input symbols for multiplesignal copies with the equalizer coefficients.
 8. The apparatus of claim1, wherein the at least one processor is operative to processmultiple-input multiple-output (MIMO) signals.
 9. The method of claim 4,further comprising: receiving multiple-input multiple-output (MIMO)signal components.
 10. The apparatus of claim 6, further comprising:means for receiving multiple-input multiple-output (MIMO) signalcomponents.
 11. The method of claim 4, further comprising: obtaininginput samples from the received signal at twice the received signal'schip rate; transforming the input samples to frequency domain using aK-point Fourier transform to yield a lower copy of the received signal'sspectrum and an upper copy of the received signal's spectrum, whereinthe lower copy comprises frequency bin indices 1 through K/2, and theupper copy comprises frequency bin indices K/2+1 through K, where K isan integer greater than or equal to
 1. 12. The method of claim 11,further comprising: selecting as the first frequency bin k₁ a frequencybin for which there are non-negligible signal components on bothfrequency bin k₁ in the lower copy and frequency bin k₁+K/2 on the uppercopy; and selecting as the disparate second frequency bin k₂ a frequencybin for which there are non-negligible signal components on only one ofeither frequency bin k₂ on the lower copy or frequency bin k₂+K/2 on theupper copy.
 13. The apparatus of claim 6, further comprising: means forobtaining input samples from the received signal at twice the receivedsignal's chip rate; and means for transforming the input samples tofrequency domain using a K-point Fourier transform to yield a lower copyof the received signal's spectrum and an upper copy of the receivedsignal's spectrum, wherein the lower copy comprises frequency binindices 1 through K/2, and the upper copy comprises frequency binindices K/2+1 through K, where K is an integer equal to or greater thanone.
 14. The apparatus of claim 13, further comprising: means forselecting as the first frequency bin k₁ a frequency bin for which thereare non-negligible signal components on both frequency bin k₁ in thelower copy and frequency bin k₁+K/2 on the upper copy; and means forselecting as the disparate second frequency bin k₂ a frequency bin forwhich there are non-negligible signal components on only one of eitherfrequency bin k₂ on the lower copy or frequency bin k₂+K/2 on the uppercopy.
 15. An apparatus comprising: at least one processor configured toderive equalizer coefficients for multiple signal copies obtained viamultiple receive antennas and oversampling, and to filter input symbolsfor the multiple signal copies with the equalizer coefficients, whereinthe at least one processor derives at least one space-frequencyequalizer for a first set of frequency bins in at least one of themultiple signal copies and derives at least one spatial equalizer for adifferent second set of frequency bins in the at least one of themultiple signal copies; and a memory coupled to the at least oneprocessor.
 16. The apparatus of claim 15, wherein the at least oneprocessor derives the equalizer coefficients based on minimum meansquare error (MMSE) criterion.
 17. The apparatus of claim 15, whereinthe at least one processor filters the input symbols in frequency domainwith the equalizer coefficients.
 18. The apparatus of claim 15, whereinthe at least one processor combines filtered symbols for the multiplesignal copies across space and frequency dimensions.
 19. An apparatuscomprising: a computer-readable medium having stored thereoncomputer-executable instructions executable by a processor to: deriveequalizer coefficients for multiple signal copies obtained via multiplereceive antennas and oversampling, the deriving comprising at least:deriving a space-frequency equalizer coefficient for a first subset offrequency bins in at least one of the multiple signal copies; andderiving a spatial equalizer coefficient for a disparate second subsetof frequency bins in the at least one of the multiple signal copies; andfiltering input symbols for the multiple signal copies with theequalizer coefficients.
 20. The apparatus of claim 19, wherein thefiltering the input symbols comprises filtering the input symbols infrequency domain with the equalizer coefficients.
 21. The apparatus ofclaim 19, wherein the filtering the input symbols comprises combiningfiltered symbols for the multiple signal copies across space andfrequency dimensions.
 22. An apparatus comprising: means for derivingequalizer coefficients for multiple signal copies obtained via multiplereceive antennas and oversampling, the means for deriving comprising atleast means for deriving a space-frequency equalizer coefficient for afirst subset of frequency bins in at least one of the multiple signalcopies and means for deriving a spatial equalizer coefficient for adisparate second subset of frequency bins in the at least one of themultiple signal copies; and means for filtering input symbols for themultiple signal copies with the equalizer coefficients.
 23. Theapparatus of claim 22, wherein the means for filtering the input symbolscomprises means for filtering the input symbols in frequency domain withthe equalizer coefficients.
 24. The apparatus of claim 22, wherein themeans for filtering the input symbols comprises means for combiningfiltered symbols for the multiple signal copies across space andfrequency dimensions.
 25. An apparatus comprising: at least oneprocessor configured to obtain input symbols for multiple (M) signalcopies from multiple (R) receive antennas and multiple (C) timesoversampling, where M is equal to R times C, to derive equalizercoefficients for the M signal copies, to filter the input symbols forthe M signal copies with the equalizer coefficients, and to combinefiltered symbols for the M signal copies, wherein R, C, and M areintegers greater than one; and a memory coupled to the at least oneprocessor, wherein the at least one processor derives M sets ofequalizer coefficients for the respective M signal copies for each of atleast one data stream, the M sets of equalizer coefficients including atleast one set comprising a space-frequency equalizer coefficient used tofilter a first subset of frequency bins in at least one of the M signalcopies and a spatial equalizer coefficient used to filter a differentsecond subset of frequency bins in the at least one of the M signalcopies.
 26. The apparatus of claim 25, wherein the at least oneprocessor receives input samples at C times chip rate for each of themultiple receive antennas and transforms the input samples for each ofthe multiple receive antennas to frequency domain to obtain inputsymbols for M signal copies from the respective receive antennas. 27.The apparatus of claim 25, wherein R is equal to two and C is equal totwo, and wherein the at least one processor derives four sets ofequalizer coefficients for four signal copies, each set of equalizercoefficients being for one of the M signal copies.
 28. The apparatus ofclaim 25, wherein the at least one processor combines signal componentson frequency bin k for the M signal copies, where k is a frequency binindex.
 29. The apparatus of claim 25, wherein the at least one processorderives the equalizer coefficients based on minimum mean square error(MMSE) criterion.
 30. The apparatus of claim 25, wherein the at leastone processor derives the equalizer coefficients based on an assumptionof uncorrelated noise for M signal copies from each of the multiplereceive antennas.
 31. The apparatus of claim 25, wherein the at leastone processor derives the equalizer coefficients based on an assumptionof uncorrelated noise for the R receive antennas.
 32. The apparatus ofclaim 25, wherein the at least one processor derives the equalizercoefficients based on an assumption of spatially and spectrallyuncorrelated noise for the M signal copies.
 33. The apparatus of claim25, wherein for each of multiple (T) data streams to be recovered, theat least one processor derives equalizer coefficients for the M signalcopies for the data stream, filters the input symbols for the M signalcopies with the equalizer coefficients to obtain filtered symbols forthe M signal copies, and combines the filtered symbols for the M signalcopies to obtain output symbols for the data stream, where T is aninteger greater than one.
 34. The apparatus of claim 33, wherein foreach of the T data streams the at least one processor derives noise andinterference covariance matrices and derives the equalizer coefficientsfor the data stream based on the noise and interference covariancematrices.
 35. The apparatus of claim 33, wherein for each of multiplefrequency bins the at least one processor derives a common noise andinterference covariance matrix and derives the equalizer coefficientsfor each of the T data streams based on the common noise andinterference covariance matrix.
 36. A method comprising: obtaining inputsymbols for multiple (M) signal copies from multiple (R) receiveantennas and multiple (C) times oversampling, where M is equal to Rtimes C; deriving equalizer coefficients for the M signal copies;filtering the input symbols for the M signal copies with the equalizercoefficients to yield filtered signals, the filtering comprising atleast filtering a first subset of frequency bins in at least one of theM signal copies using a space-frequency equalizer coefficient, andfiltering a different second subset of frequency bins in the at leastone of the M signal copies using a spatial equalizer coefficient; andcombining the filtered symbols for the M signal copies, where M, R, andC are integers greater than one.
 37. The method of claim 36, wherein thecombining the filtered symbols comprises combining signal components onfrequency bin k for the M signal copies, where k is a frequency binindex.
 38. The method of claim 36, wherein the deriving the equalizercoefficients comprises deriving the equalizer coefficients based on anassumption of uncorrelated noise for M signal copies from each receiveantenna.
 39. The method of claim 36, wherein the deriving the equalizercoefficients comprises deriving the equalizer coefficients based on anassumption of spatially and spectrally uncorrelated noise for the Msignal copies.
 40. The method of claim 36, wherein the deriving theequalizer coefficients comprises deriving the equalizer coefficients forthe M signal copies for each of at least one data stream, wherein thefiltering the input symbols comprises filtering the input symbols forthe M signal copies with the equalizer coefficients for each data streamto obtain filtered symbols for the M signal copies for the data stream,and wherein the combining the filtered symbols comprises combining thefiltered symbols for the M signal copies for each data stream to obtainoutput symbols for the data stream.